Dual R-groups of the inner forms of SL(N)

TL;DR

This paper investigates the R-groups of inner forms of SL(N) over non-archimedean fields, confirming conjectures about their structure and providing new examples of phenomena not seen in classical groups.

Contribution

It proves that the R-groups of inner forms of SL(N) are isomorphic to their duals as conjectured by Arthur, and describes the associated 2-cocycles, extending understanding of these groups.

Findings

R-groups are isomorphic to their duals as per Arthur's conjecture

Explicit description of 2-cocycles attached to R-groups

Construction of examples showing new phenomena in inner forms

Abstract

We study the Knapp-Stein R-groups of the inner forms of SL(N) over a non-archimedean local field of characteristic zero, by using restriction from the inner forms of GL(N). As conjectured by Arthur, these R-groups are then shown to be naturally isomorphic to their dual avatars defined in terms of L-parameters. The 2-cocycles attached to R-groups can be described as well. The proofs are based on the results of K. Hiraga and H. Saito. We also construct examples to illustrate some new phenomena which do not occur in the case of SL(N) or classical groups.